# Analytical formula for discounted exposure of a European Put on a stock in Real-World measure

Is there an analytical formula to approximate the discounted exposure for a European Put on a Stock in the Real-World measure? This is just an initial phase to be able to assess the accuracy of using Longstaff-Schwartz regression method, using a simple example. I would like to compare the regression results with analytical solution for discounted exposure for a European Put on a Stock.

Also, is it fine to calculate the expected stock price at future points as $$E[S(T_{k})] = S(T_{0}) * exp (\mu * T_{k})$$ where $$T_{k}$$ are future time points for $$k = 1, 2, .. , M$$, with $$\mu$$ being the real-world drift of the stock, and subsequently, use Black-Scholes analytical formula for valuing a put using $$S(T_{k})$$ calculated above - this is in order to calculate the approximate expected exposure at a future time point, $$t_{k}$$?

Thanks in advance for any insight into this.

• Please use MathJax for formatting in the future. Jul 20, 2023 at 12:16
• How do you define discounted future exposure. Jul 20, 2023 at 17:25
• Calculate exposure as conditional expected value at a future time point, t, and then discount it to today, say, t0. Jul 21, 2023 at 9:18

The put price at any point in the future $$t$$, with stock price $$S(t)$$, is just $$BS(T-t,S(t),K)$$, i.e. the black-scholes price. In american options context the "continuation value" is always the black-scholes price. This is the value I would want the longstaff algorithm to be able to approximate.

The expected stock price at any point in the future $$t$$ is $$S(t)=S(0)*exp(mu+vol^2/2)$$.

To answer your comment, "can I use the expected stock price at a future time t to plug into BS to get the put price at this future time, t? Is it correct to do so?"

You need $$E(BS(T-t,S(t))$$ which is different than $$BS(T-t,E(S(t))$$.
• Thank you, yes, but to use $BS(T−t,S(t),K)$, I would need to simulate S(t). My goal is to have an approximation using just closed form solution - is it possible, especially, in the real-world-measure? My question was, can I use the expected stock price at a future time t to plug into BS to get the put price at this future time, t? Is it correct to do so? Again, this is only to have an approximation i.e., a guide to use as a benchmark. Jul 21, 2023 at 9:16